Question

exterior angles of polygons
question
the measures of the exterior angles of an octagon are ( x ^ { circ }, 2 x ^ { circ }, 3 x ^ { circ }, 4 x ^ { circ }, 5 x ^ { circ }, 6 x ^ { circ }, 7 x ^ { circ } ), and ( 8 x ^ { circ } ). solve for ( x ).
answer
attempt 1 out of 2
( x = )

Explanation:

Step1: Sum of exterior angles of a polygon

The sum of the exterior angles of any polygon is \(360^{\circ}\).

Step2: Set up the equation

We have the exterior angles \(x^{\circ},2x^{\circ},3x^{\circ},4x^{\circ},5x^{\circ},6x^{\circ},7x^{\circ},8x^{\circ}\).
Sum them up: \(x + 2x+3x + 4x+5x+6x+7x+8x=360\).
Combine like - terms: \((1 + 2+3 + 4+5+6+7+8)x=360\).
Since \(1+2+3+\cdots+8=\frac{8\times(8 + 1)}{2}=36\) (using the formula for the sum of the first \(n\) positive integers \(S_n=\frac{n(n + 1)}{2}\), here \(n = 8\)).
The equation becomes \(36x=360\).

Step3: Solve for \(x\)

Divide both sides of the equation \(36x=360\) by \(36\): \(x=\frac{360}{36}\).

Answer:

\(x = 10\)